Lebesgue measure of escaping sets of entire functions

Abstract

For a transcendental entire function $f$ of finite order in the Eremenko-Lyubich class $\mathcal{B}$, we give conditions under which the Lebesgue measure of the escaping set $\mathcal{I}(f)$ of $f$ is zero. This is inspired by the recent work of Aspenberg and Bergweiler, in which they give conditions on entire functions in the same class with escaping sets of positive Lebesgue measure. We will construct an entire function in the Eremenko-Lyubich class to show that the condition given by Aspenberg and Bergweiler is essentially sharp. Furthermore, we adapt our idea of proof to the case of infinite order entire functions. Under some restrictions to the growth of these entire functions, we show that the escaping sets have zero Lebesgue measure. This generalizes a result of Eremenko and Lyubich.

30 pages. Theorem 1.2 added. Some overall revision

References in corpus (4)

• Dynamics of meromorphic functions with direct or logarithmic singularities
• Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture
• Hyperbolic entire functions and the Eremenko-Lyubich class: Class $\mathcal{B}$ or not class $\mathcal{B}$?
• Lebesgue measure of escaping sets of entire functions of completely regular growth

Cited by in corpus (4)

• The escaping set in transcendental dynamics
• Hausdorff dimension of escaping sets of meromorphic functions
• Speiser meets Misiurewicz
• Hausdorff dimension of escaping sets of Nevanlinna functions